The equation \( p=1.65 t^{2}+18.25 t+155 \) approximates the average sale price \( p \) of a house (in thousands of dollars) for years \( t \) since 2010. What is the best estimate for the price of the house in year 2018 ? \( \begin{array}{l}\text { W } \$ 364,000 \\ \$ 407,000 \\ \$ 427,000\end{array} \) \( \begin{array}{l}\text { W453,000 }\end{array} \)

To find the best estimate for the price of the house in the year 2018, we need to determine \( t \). Since 2018 is 8 years after 2010, we set \( t = 8 \). Plugging this into the equation gives us: \[ p = 1.65(8^2) + 18.25(8) + 155 \] Calculating this step-by-step: 1. \( 8^2 = 64 \) 2. \( 1.65 \times 64 = 105.6 \) 3. \( 18.25 \times 8 = 146 \) 4. Finally, adding: \( 105.6 + 146 + 155 = 406.6 \) Rounding this to the nearest thousand gives us \( p \approx 407 \). Thus, the best estimate for the price of the house in 2018 is **$407,000**. The formula offers a glimpse into the evolution of real estate prices, which often reflect economic trends, demographic changes, and infrastructure development over the years. By looking back, one can see how external factors like market demand and interest rates play into these numbers, often leading to surprising insights about housing affordability and investment potential. For those keen on mastering these equations, consider exploring more about the role of quadratic equations in predicting trends. They pop up in various fields, from physics modeling projectile motion to calculating profit in business strategies. Sites like Khan Academy or Math is Fun can serve as terrific resources for deeper dives into quadratic functions and their myriad applications.